Calculate the center of mass for 2D or 3D point mass systems with weighted averages, moment of inertia, and position diagrams.
The center of mass is the unique point where the weighted average of all mass positions in a system balances perfectly. For any collection of point masses, the center of mass coordinates are found by computing the mass-weighted average of each coordinate: X_cm = Σ(m·x)/M, where M is the total mass. This concept is fundamental in physics, engineering, and applied mathematics.
In two dimensions, the center of mass determines where a flat object would balance on a pin. In three dimensions, it defines the point about which a free body would rotate, and it is the point that follows a simple parabolic trajectory when the body is thrown. Calculating the center of mass is essential for structural engineering, robotics, spacecraft attitude control, and collision physics.
This calculator handles any number of point masses in both 2D and 3D. Enter each particle's mass and coordinates, then instantly see the center of mass, total mass, moment of inertia about the center, and distance of each particle from the center. The mass distribution diagram helps you visualize relative contributions.
Presets include classic configurations: triangle masses, Earth-Moon system, L-shaped distributions, 3D tetrahedra, and weighted bars. You can add or remove particles freely to model any system you need.
Whether you're solving a physics homework problem, designing a balanced mechanism, or analyzing astronomical systems, this tool provides fast, accurate center-of-mass calculations.
Finding the center of mass of a multi-particle system means summing mass-weighted coordinates across every particle in 1D, 2D, or 3D — a process that scales linearly with particle count and is a magnet for arithmetic slips. This calculator handles up to dozens of particles, computes the center of mass in all three dimensions, shows the total mass and individual weight fractions, and displays a coordinate diagram so you can visually verify the result. Physics students use it for homework checks, and engineers use it when balancing loads, positioning robotic grippers, or computing moments of inertia for composite bodies.
X_cm = Σ(m_i · x_i) / Σ(m_i); Y_cm = Σ(m_i · y_i) / Σ(m_i); Z_cm = Σ(m_i · z_i) / Σ(m_i); I = Σ m_i · r_i²
Result: CM ≈ (3.30, 1.90), total mass = 10
Three 2D point masses at (1,2), (4,1), (6,5) with masses 3, 5, 2 yield a center of mass at approximately (3.30, 1.90).
For a finite set of point masses, the center of mass is the mass-weighted average of positions: x̄ = Σmᵢxᵢ / Σmᵢ. For a continuous body with density ρ(r), the sum becomes an integral: x̄ = ∫x ρ dV / ∫ρ dV. This calculator handles the discrete case, but the same principle underlies continuous calculations. Symmetry shortcuts are powerful: the center of mass of a uniform sphere is at its geometric center, and for any body with a plane of symmetry, the center of mass lies on that plane. Decomposing complex shapes into simpler sub-bodies whose centers of mass are known lets you apply the discrete formula to the sub-body centers.
Newton's second law for a system of particles simplifies when expressed about the center of mass: F_ext = M a_cm. Internal forces cancel in pairs, so only external forces accelerate the center of mass. This is why a spinning wrench thrown in the air traces a smooth parabolic path at its center of mass even while tumbling. In the center-of-mass reference frame, the total momentum is always zero, which greatly simplifies collision and scattering calculations in both classical and relativistic mechanics.
In structural engineering, the centroid of a cross-section (the center of mass of its area) determines where axial loads produce no bending. In aerospace, the center of gravity (CG) position relative to the center of lift determines longitudinal stability — an aircraft whose CG is too far aft becomes unstable. In robotics, computing the center of mass of a manipulator arm at each joint configuration is essential for torque planning and dynamic balancing. Even in everyday life, stacking objects or balancing a seesaw is an intuitive application of center-of-mass principles.
The center of mass is the average position of all mass in a system, weighted by each mass value. It is the point where the system would balance.
In a uniform gravitational field they are identical. They differ only when gravity varies significantly across the object (e.g., very large structures).
Yes. For hollow or concave shapes (like a ring or L-bracket), the center of mass can lie in empty space.
Moment of inertia I = Σ m·r² measures rotational resistance about an axis. This calculator computes it about the center of mass.
Click the "+ Add Particle" button. You can add as many point masses as you need, and remove them individually.
This calculator handles discrete point masses. For continuous bodies, approximate them with many small mass elements or use integral formulations.